What actually happens just after a rider pushes the left grip forward to begin leaning a motorcycle left?

Answer: Front wheel goes right

Rider leans left first — The rider's body may move left as the bike begins to lean, but that is not the first mechanical step asked here. The prompt starts with a handlebar press, and that input first turns the front wheel the other way. The reliable trigger is a steering input that starts the lean.

Front wheel goes left — This is the intuitive answer, but it has the direction backward. To begin a left lean at speed, the front wheel first points slightly right, not left. That tiny wrong-way move is what shifts the tire contact path and lets the bike fall left.

Front wheel goes right ✓ — Correct. Pressing the left grip briefly steers the front wheel right, even though the rider wants a left lean. That tiny wrong-way steering input starts the lean, and the motorcycle then falls into the left side of the turn. The rider then lets the front end settle toward the curve.